the golden angle is the smaller of the two angles created by dividing a full turn—360° or 2π radians—according to the golden ratio. This means that the ratio of the full turn with the larger angle is equal to the ratio of the larger angle with the smaller angle. The golden angle is approximately equal to 137.5° or 2.4 radians.

A golden rhombus is a rhombus whose two diagonals are in the golden ratio. This property also implies that a golden rhombus is the dual polygon of a golden rectangle.

Golden rhombuses should not be confused with the two types of rhombuses found in Penrose tiles.

A Kepler triangle is a right triangle whose side lengths are in a geometric progression, that is, the ratio between the two legs is the same as the ratio between the longer leg and the hypotenuse. It should not be a surprise that the common ratio of this progression is related to the golden ratio and is exactly $\sqrt{𝜙}$.

Proof that the common ratio is $\sqrt{𝜙}$

To find the common ratio of the triangle’s side’s geometric progression, we can use the Pythagorean theorem. By setting the shorter leg to have a length of $1$, and the longer leg to have a length of $x$, the ratio we are looking for is also just $x$. The triangle’s hypotenuse is therefore $\sqrt{1^2+x^2}$.

Since the side lengths need to be in a geometric progression, we have the following equation expressing that the ratios of the longer to the shorter leg and the hypotenuse to the longer leg are the same:

$\frac{x}{1}=\frac{\sqrt{1^2+x^2}}{\mathrm{x}}$

Multiplying both sides by $x$ and then squaring both sides, we have:

$x^4=1+x^2$

Substituting $x^2$ with $𝜙$ results in the familiar golden quadratic equation. This means that $x$, the common ratio we are looking for, is the square root of the golden ratio.

Etymology

The Kepler triangle is named after Johannes Kepler because he was the first known to describe this special right triangle and its relationship to the golden ratio. The Kepler triangle also combines two things in geometry that Kepler declares as “great treasures”: the Pythagorean theorem and the golden ratio.

“Geometry has two great treasures: one is the Theorem of Pythagoras, the other the division of a line in extreme and mean ratio. The first we can compare to a mass of gold; the other we may call a precious jewel.” —Johannes Kepler

“Geometry has two great treasures: one is the Theorem of Pythagoras, the other the division of a line in extreme and mean ratio. The first we can compare to a mass of gold; the other we may call a precious jewel.” —Mysterium Cosmographicum, Johannes Kepler

Johannes Kepler (1571–1630) was a German mathematician and astronomer best known for his laws of planetary motion. As a mathematician, Kepler studied the golden ratio—which he referred to as the extreme and mean ratio—and has compared it to a precious jewel, which inspired this website’s name. He declared the golden ratio together with the Pythagorean theorem as geometry’s “two great treasures”. Ironically, Kepler compared the Pythagorean theorem to a “mass of gold”, which would have been more appropriate for the extreme and mean ratio since this is now popularly known as the golden ratio.